Chapter 2 Elements of Abstract Group Theory

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20 Elements of Abstract Group Theory which is important, not any particular realization of the group. Further discussion of this point will be taken up in the next chapter. 2.3 Elementary Properties of Groups The examples in the preceding section showed that all groups are endowed with several general properties. In this section, we deduce some additional properties which, although evident in particular examples, can be shown generally to follow from the properties of abstract groups. Theorem 2.1. (Uniqueness of the identity) The identity element in a group G is unique. Proof. Suppose there are two identity elements e and e ′ in G. Then, according to the definition of a group, we must have that and ae = a e ′ a = a for all a in G. Setting a = e ′ in the first of these equations and a = e in the second shows that so e = e ′ . e ′ = e ′ e = e, This theorem enables us to speak of the identity e of a group. The notation e is derived from the German word Einheit for unity. Another property common to all groups is the cancellation of common factors within equations. This property owes its existence to the associativity of the group composition rule.
Elements of Abstract Group Theory 21 Theorem 2.2. (Cancellation) In a group G, the left and right cancellation laws hold, i.e., ab = ac implies b = c and ba = ca implies b = c. Proof. Suppose that ab = ac. Let a −1 be an inverse of a. Then, by left-multiplying by this inverse, and invoking associativity, we obtain a −1 (ab) =a −1 (ac) (a −1 a)b =(a −1 a)c, eb = ec , so b = c. Similarly, beginning with ba = ca and right-multiplying by a −1 shows that b = c in this case also. Notice that the proof of this theorem does not require the inverse of a group element to be unique; only the existence of an inverse was required. In fact, the cancellation theorem can be used to prove that inverses are, indeed, unique. Theorem 2.3. (Uniqueness of inverses) For each element a in a group G, there is a unique element b in G such that ab = ba = e. Proof. Suppose that there are two inverses b and c of a. Then ab = e and ac = e. Thus, ab = ac, so by the Cancellation Theorem, b = c. As in the case of the identity of a group, we may now speak of the inverse of every element in a group, which we denote by a −1 . As was discussed in Sec. 2.1, this notation is borrowed from ordinary multiplication, as are most other notations for the group composition rule. For example, the n-fold product of a group element g with itself is denoted
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Chapter 2 Elements of Abstract Group Theory

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